Atoms are excited to an upper excited state depending on the wavelength of excitation source or laser and their transition probability. All the excited atoms will get de-excited to various other lower levels, where transition probability is allowed. Finally, one can get different peaks corresponding to each transition.

If you consider both absorption and radiation is involved when light interacts with matter, then you can think of Luminescence process be described as the inverse of absorption, inform of a radiative process created by the recombination of excited electrons (as supposed the non-radiative recombination processes leading to thermal effects). Therefore, one may observe transitions between band-edges (band-to-band transition) or in presence of impurities for doped semiconductors one may observe lines associated with transition from within the conduction band (donor states) and valence band (acceptor states). Most likely multiple lines observed in your experiment indicates transitions involving impurity states. The exact role of excitation wavelength is that to observe all possible excitations, the excitation wavelength should be shorter than the most energetic transition, otherwise some if not all transitions will be absent in the PL spectra. The following article shows that PL spectra of doped nanocrystal include emission peaks at higher wavelengths compared to the un-doped ZnS nanoparticles: https://www.sciencedirect.com/science/article/pii/S0749603615002499,

Also, the link below leads to a more general description of PL spectroscopy: https://www.sciencedirect.com/topics/chemistry/photoluminescence-spectroscopy